Young's Formula

Date: 08/21/99 at 20:33:53
From: Eric Smith
Subject: Problems with units
I'm having trouble finding information on how to solve problems with
units, such as physics or chemistry problems. Specifically, I don't
know how to solve "Young's formula" for calculating a child's dose of
medicine based on the adult dose.
C = A( g / g + 12)
where
A = adult dose in mg
g = child's age
For example, if adult dose is 600 mg and child's age is 3 years, what
is C? This sounds so easy, but it's driving me crazy.
C = 600 mg (3 yr / 3 yr + 12)
C = 1800 mg * yr / 3 yr + 12
Now what?
Thanks.

Date: 08/22/99 at 23:27:16
From: Doctor Peterson
Subject: Re: Problems with units
Hi, Eric. Surprisingly, this is a fascinating question, because I can
see how you can get hung up on this even though it really is a simple
problem. You're being too conscientious, in a sense, and you have hit
upon a conflict between two ways of presenting formulas that I hadn't
thought about before.
It looks like you've learned to carry the units along in the
calculation, which is a great method that I like to use. You just have
to be careful that the formula is set up right before you use this
method, something I don't think I've ever heard taught.
First, we have to be clear what the formula is. Parentheses would have
been helpful. Here's what I think it should be:
g
C = A * ------
g + 12
Second, because there is an unlabeled constant in the formula, before
we can replace the variables with labeled values, we need to label the
constant. The formula was written with a different convention in mind,
namely that each variable is expected to be a number representing a
_fixed unit_; I would write it carefully like this:
g
C = A * ------
g + 12
where
A = adult dose in mg
g = child's age in years
C = child's dose in mg
To work the problem this way, as intended, you simply plug in the
numbers and apply the specified unit at the end:
For A = 600 and g = 3:
3 3
C = 600 * ------ = 600 * -- = 120 mg
3 + 12 15
To work this problem with units attached to the numbers, we have to
recognize that since g is assumed to be in years, 12 must be in years:
g
C = A * ---------
g + 12 yr
where
A = adult dose
g = child's age
C = child's dose
Written this way, we could use any units for A and g, and will get an
answer for C that is correct in whatever units it ends up with,
including any conversions you want to do:
For A = 600 mg and g = 3 yr:
3 yr 3 yr
C = 600 mg * ------------ = 600 mg * ----- = 120 mg
3 yr + 12 yr 15 yr
If g were in days, you would find it necessary to convert it to years
before adding; if A were in pounds, you would get C in pounds.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/